This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the PC based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality of high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

翻译：本文提出了随机虚拟单元方法，用于传播线性弹性随机问题中的不确定性。我们首先推导了适用于二维和三维线性弹性问题的随机虚拟单元方程，这些问题可能涉及材料属性、外力等的不确定性。为此，我们构建了一个耦合确定性虚拟单元空间与随机空间的随机虚拟单元空间，并用于近似未知的随机解。随后，开发了两种数值框架来求解所推导的随机虚拟单元方程，包括基于多项式混沌近似的方法和基于弱侵入近似的方法。在基于PC的框架中，随机解使用多项式混沌基进行近似，并通过应用随机伽辽金过程于原始随机虚拟单元方程生成的增广确定性虚拟单元方程来求解。在基于弱侵入近似的框架中，随机解表示为随机变量与确定性向量乘积的求和，其中确定性向量通过随机伽辽金方法将原始随机问题转化为确定性虚拟单元方程求解，而随机变量则通过经典伽辽金过程将原始随机问题转化为一维随机代数方程求解。该方法成功避免了高维随机问题的维数灾难，因为所有随机输入均嵌入到计算量弱依赖于随机维度的一维随机代数方程中。针对低维和高维随机输入的二维与三维问题的数值结果，展示了所提方法的良好性能。